A210877 Triangle of coefficients of polynomials v(n,x) jointly generated with A210876; see the Formula section.

1, 0, 3, 0, 3, 4, 0, 2, 8, 5, 0, 2, 6, 17, 6, 0, 2, 5, 18, 31, 7, 0, 2, 5, 14, 47, 51, 8, 0, 2, 5, 13, 41, 107, 78, 9, 0, 2, 5, 13, 35, 115, 218, 113, 10, 0, 2, 5, 13, 34, 98, 296, 407, 157, 11, 0, 2, 5, 13, 34, 90, 276, 695, 709, 211, 12, 0, 2, 5, 13, 34, 89, 244, 750

Offset

1,3

Comments

For n>2, each row begins with 0 and ends with n+1. If the term in row n and column k is denoted by U(n,k), then U(n,n-2)=A105163(n-1).

Row sums: A000225 (-1+2^n)

Alternating row sums: A137470

Limiting row: 0,2,5,13,34,89,..., even-indexed Fibonacci numbers

For a discussion and guide to related arrays, see A208510.

Links

Table of n, a(n) for n=1..74.

Formula

u(n,x)=x*u(n-1,x)+v(n-1,x)+1,

v(n,x)=x*u(n-1,x)+x*v(n-1,x)+x,

where u(1,x)=1, v(1,x)=1.

Example

First six rows:
1
1...2
1...1...3
1...1...3...4
1...1...2...8...5
1...1...2...6...17...6
First three polynomials v(n,x): 1, 1 + 2x, 1 + x + 3x^2

Mathematica

u[1, x_] := 1; v[1, x_] := 1; z = 14;
u[n_, x_] := x*u[n - 1, x] + v[n - 1, x] + 1;
v[n_, x_] := x*u[n - 1, x] + x*v[n - 1, x] + x;
Table[Expand[u[n, x]], {n, 1, z/2}]
Table[Expand[v[n, x]], {n, 1, z/2}]
cu = Table[CoefficientList[u[n, x], x], {n, 1, z}];
TableForm[cu]
Flatten[%]    (* A210876 *)
cv = Table[CoefficientList[v[n, x], x], {n, 1, z}];
TableForm[cv]
Flatten[%]    (* A210877 *)
Table[u[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[v[n, x] /. x -> 1, {n, 1, z}]  (* A000225 *)
Table[u[n, x] /. x -> -1, {n, 1, z}] (* A077973 *)
Table[v[n, x] /. x -> -1, {n, 1, z}] (* A137470 *)

Crossrefs

Cf. A210876, A208510.

Sequence in context: A210485 A111815 A281269 * A127753 A197736 A073367

Adjacent sequences: A210874 A210875 A210876 * A210878 A210879 A210880

Keyword

nonn,tabl

Author

Clark Kimberling, Mar 30 2012

Status

approved